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PROFESSOR: Hi.
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Today we embark on a
new phase of our course,
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signified by the
beginning of block two.
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Where in block one,
after indicating what
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mathematical structure
was, in our quest
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to get as meaningfully as
possible into the question
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of functions of
several variables,
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we introduced the
study of vectors.
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Vectors were important
in their own right,
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so we paid some attention
to that particular topic.
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And now we're at the next
plateau, where one introduces
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the concept of functions.
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In other words, just like
in ordinary arithmetic,
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after we get through with the
arithmetic of constants, which
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is what you could basically
call elementary school
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arithmetic-- you work
with fixed numbers--
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we then move into algebra.
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And from algebra we
move into calculus.
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Sooner or later, we would like
to study the calculus of vector
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functions and the like.
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And why not sooner?
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Which is what brings
us to today's lesson.
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We are going to talk about
a rather difficult mouthful,
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I guess.
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I call today's lecture "Vector
Functions of Scalar Variables."
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And if that sounds
like a tongue twister,
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let me point out that what has
happened now is the following.
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When we did part
one of this course,
00:02:10.479 --> 00:02:13.580
when we talked about functions
and function machines,
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recall that both the input
and the output of our machine
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were, by definition, scalars--
in other words, real numbers.
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Remember, we talked
about functions
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of a single real variable.
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Now we have at our disposal
both vectors and scalars.
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Consequently, the input can be
either a vector or a scalar,
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and the output can be
either a vector or a scalar.
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And this gives us an
almost endless number
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of ways of combining these,
where by "almost endless,"
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I mean four.
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And the reason I say
"almost endless" is this.
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There are only four ways.
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What are the four ways?
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Well, the four ways are what?
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Imagine that x is a scalar.
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In other words, the
input is a scalar.
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Then the two
possibilities are what?
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That the output is either
a scalar or a vector.
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On the other hand,
the input could
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have been a vector, in which
case, there would have been
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two additional possibilities.
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Namely, the output would be
either a scalar or a vector.
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Now the reason I call these
four possibilities endless,
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or nearly endless, is
the fact that by the time
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we get through discussing
all four possible cases,
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the course will be
essentially over.
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And to give you some
hindsight as to what I mean,
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remember that we had a
rather long course that
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was part one of this course.
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And notice that part
one of this course
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was concerned only with
that one case in four
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where both the
input and the output
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happen to be real
numbers, scalars.
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And it took us that long
to develop the subject
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called what?
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Real functions of a single
real variable-- input
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a scalar, output a scalar.
00:03:52.150 --> 00:03:57.610
Now the one that we've
chosen for our next block,
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next combination
that we've chosen,
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is where the input will
be a scalar and the output
00:04:02.790 --> 00:04:05.980
will a vector.
00:04:05.980 --> 00:04:08.010
And in terms of
physical examples,
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this can be made
very, very meaningful.
00:04:10.630 --> 00:04:14.050
For example, one very
common physical situation
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is that when we study
force on an object-- force
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is obviously a vector-- but
that the force on an object
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usually varies with time.
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But time happens to be a scalar.
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In other words, we
might, for example,
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write that the vector F is
a function of the scalar t.
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And rather than talk
abstractly, let me make up
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a pseudo-physical situation.
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By pseudo-physical,
I mean I haven't
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the vaguest notion where
this formula would come up
00:04:44.160 --> 00:04:45.380
in real life.
00:04:45.380 --> 00:04:47.590
And I think that after
you saw my performance
00:04:47.590 --> 00:04:51.000
on the explanation of work
equals force times distance
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and why objects don't rise
under the influence of friction
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and what have you,
you believe me
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when I tell you I have no feel
for these physical things.
00:04:59.740 --> 00:05:01.320
But I say that semi-jokingly.
00:05:01.320 --> 00:05:04.020
The main reason is that
it's irrelevant what
00:05:04.020 --> 00:05:05.990
the physical application is.
00:05:05.990 --> 00:05:07.630
The important point
is that each of you
00:05:07.630 --> 00:05:10.250
will find physical
applications in your own way.
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All we really need is
some generic problem
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that somehow signifies what
all other problems look like.
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What I'm driving
at here is, imagine
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that we have a force
F, which varies
00:05:20.790 --> 00:05:24.710
with time, written in Cartesian
coordinates as follows.
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The force is e to the
minus t times i plus j.
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Notice that t is a scalar.
00:05:31.890 --> 00:05:36.380
As t varies, the scalar
e to the minus t varies.
00:05:36.380 --> 00:05:39.110
But the scalar e to
the minus t varying
00:05:39.110 --> 00:05:43.920
means that the vector e to
the minus t i is changing,
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is varying.
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In other words, notice that
e to the minus t i plus j
00:05:49.520 --> 00:05:53.870
is a variable vector,
even though t is a scalar.
00:05:53.870 --> 00:05:57.300
For example, when t happens
to be 0, what is the force?
00:05:57.300 --> 00:05:59.710
When t is 0, this
would read what?
00:05:59.710 --> 00:06:03.550
The force is e to
the minus 0 i plus j.
00:06:03.550 --> 00:06:05.890
That's the same as saying--
since e to the minus 0
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is 1-- the force,
when the time is 0,
00:06:08.950 --> 00:06:11.890
is i plus j, which is a vector.
00:06:11.890 --> 00:06:14.540
You see, that vector
changes as t changes,
00:06:14.540 --> 00:06:17.530
but t happens to be a scalar.
00:06:17.530 --> 00:06:21.030
Now here's the interesting
point or an interesting point.
00:06:21.030 --> 00:06:23.310
I look at this expression,
and I'm tempted
00:06:23.310 --> 00:06:25.490
to say something like this.
00:06:25.490 --> 00:06:31.310
I say for large values of t,
F of t is approximately j.
00:06:31.310 --> 00:06:33.120
Now how did I arrive at that?
00:06:33.120 --> 00:06:34.890
Well what I said was,
is I said, you know,
00:06:34.890 --> 00:06:39.040
e to the minus t gets very
close to 0 as t gets large.
00:06:39.040 --> 00:06:44.500
As t gets large, therefore,
this component tends to 0.
00:06:44.500 --> 00:06:48.970
This component stays constantly
j-- or the component is 1,
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the vector is j.
00:06:50.860 --> 00:06:55.250
So that as t gets large,
F of t behaves like j.
00:06:55.250 --> 00:06:58.930
And what I'm leading up
to is that intuitively,
00:06:58.930 --> 00:07:01.040
I am now using
the limit concept.
00:07:01.040 --> 00:07:02.560
I'm saying to myself, what?
00:07:02.560 --> 00:07:05.110
The limit of F of
t as t approaches
00:07:05.110 --> 00:07:08.050
infinity et cetera is
the limit of this sum.
00:07:08.050 --> 00:07:10.260
The limit of the sum is
the sum of the limits.
00:07:10.260 --> 00:07:12.680
And I now take the
limit term by term.
00:07:12.680 --> 00:07:16.870
And I sense that this limit
is 0 and that this one is 1.
00:07:16.870 --> 00:07:18.920
Now you see, the
whole point is this.
00:07:18.920 --> 00:07:23.830
What I would like to do is
to study vector calculus.
00:07:23.830 --> 00:07:25.010
Meaning in this case, what?
00:07:25.010 --> 00:07:30.200
The calculus of vector
functions of a scalar variable,
00:07:30.200 --> 00:07:32.180
of a scalar input.
00:07:32.180 --> 00:07:36.610
Now here's why structure was
so important in our course.
00:07:36.610 --> 00:07:39.600
What I already know how
to do is study limits
00:07:39.600 --> 00:07:43.320
in the case where both my input
and my output were scalars.
00:07:43.320 --> 00:07:45.180
What I would like
to be able to do
00:07:45.180 --> 00:07:49.790
is structurally inherit
that entire system.
00:07:49.790 --> 00:07:51.150
Because it's a beautiful system.
00:07:51.150 --> 00:07:52.460
I understand it well.
00:07:52.460 --> 00:07:54.640
I have a whole bunch of
theorems in that system.
00:07:54.640 --> 00:07:58.270
If I could only incorporate
that verbatim into vector
00:07:58.270 --> 00:08:01.540
calculus, not only would
I have a unifying thread,
00:08:01.540 --> 00:08:05.930
but I can save weeks of work
not having to re-derive theorems
00:08:05.930 --> 00:08:09.339
for vectors which automatically
have to be true because
00:08:09.339 --> 00:08:10.130
of their structure.
00:08:10.130 --> 00:08:12.980
Now that's, again, a
difficult mouthful.
00:08:12.980 --> 00:08:15.770
This is written up
voluminously in our notes.
00:08:15.770 --> 00:08:17.970
It's emphasized
in the exercises,
00:08:17.970 --> 00:08:21.040
but I thought that I would try
to show you a little bit live
00:08:21.040 --> 00:08:22.580
what this thing really means.
00:08:22.580 --> 00:08:24.330
Because I think that
somehow or other,
00:08:24.330 --> 00:08:28.680
you have to hear these ideas,
rather than just read them,
00:08:28.680 --> 00:08:30.640
to get the true feeling.
00:08:30.640 --> 00:08:33.919
What we do, for example,
is we re-visit limits.
00:08:33.919 --> 00:08:36.200
And we write down the
definition of limit
00:08:36.200 --> 00:08:39.409
just as it appeared
in the scalar case.
00:08:39.409 --> 00:08:41.950
And let's start with an
intuitive approach first,
00:08:41.950 --> 00:08:44.890
and later on, we'll get to
the epsilon-delta approach.
00:08:44.890 --> 00:08:47.460
We said that the limit
of f of x as x approaches
00:08:47.460 --> 00:08:53.090
a equals L means
that f of x is near L
00:08:53.090 --> 00:08:55.690
provided that x is "near" a.
00:08:55.690 --> 00:08:58.930
And I've written the word
"near" in quotation marks here
00:08:58.930 --> 00:09:02.632
to emphasize that we were using
a geometrical phrase-- well,
00:09:02.632 --> 00:09:04.340
I guess you can't call
one word a phrase,
00:09:04.340 --> 00:09:08.180
but a geometrical term--
to emphasize an arithmetic
00:09:08.180 --> 00:09:08.720
concept.
00:09:08.720 --> 00:09:13.530
Namely, to say that f of x
was near L where f of x and L
00:09:13.530 --> 00:09:17.200
were numbers meant
quite simply that what?
00:09:17.200 --> 00:09:20.710
The numerical value of f
of x was very nearly equal
00:09:20.710 --> 00:09:24.110
to the numerical value of L. And
that's the same as saying what?
00:09:24.110 --> 00:09:28.740
That the absolute value of
the difference f of x minus L
00:09:28.740 --> 00:09:31.010
was very small.
00:09:31.010 --> 00:09:32.830
I just mention that, OK?
00:09:32.830 --> 00:09:37.110
Now what we would like to do
is invent a definition of limit
00:09:37.110 --> 00:09:40.400
for a vector-valued function
of a scalar variable.
00:09:40.400 --> 00:09:42.400
One way of doing
this is of course
00:09:42.400 --> 00:09:45.240
to make up a completely
brand new definition that has
00:09:45.240 --> 00:09:47.160
nothing to do with the past.
00:09:47.160 --> 00:09:49.380
But as in every field
of human endeavor,
00:09:49.380 --> 00:09:54.480
one likes to plan the
future and further sorties
00:09:54.480 --> 00:09:58.530
after one has modeled the
successes of the past.
00:09:58.530 --> 00:10:01.210
So what we do is a
very simple device
00:10:01.210 --> 00:10:02.980
and yet very, very powerful.
00:10:02.980 --> 00:10:07.340
We vectorize the definition that
has already served us so well.
00:10:07.340 --> 00:10:09.740
What I mean by
that is, I go back
00:10:09.740 --> 00:10:12.800
to the definition which I've
written here and put in vectors
00:10:12.800 --> 00:10:14.260
in appropriate places.
00:10:14.260 --> 00:10:17.490
For example, we're dealing
with a vector function,
00:10:17.490 --> 00:10:20.040
so f has to have
an arrow over it.
00:10:20.040 --> 00:10:23.130
Moreover, since f
of x is a vector,
00:10:23.130 --> 00:10:26.100
anything that it approaches
as a limit by definition
00:10:26.100 --> 00:10:27.860
must also be a vector.
00:10:27.860 --> 00:10:31.490
So that means the L
must also be arrowized.
00:10:31.490 --> 00:10:32.810
All right?
00:10:32.810 --> 00:10:34.710
Put an arrow over the L.
00:10:34.710 --> 00:10:36.660
On the other hand,
the x and the a
00:10:36.660 --> 00:10:40.890
remain left alone because after
all, they are just scalars.
00:10:40.890 --> 00:10:46.010
Remember again, x and a are
inputs of our function machine.
00:10:46.010 --> 00:10:47.560
And in the particular
investigation
00:10:47.560 --> 00:10:50.430
that we're making, our input
happens to be a scalar.
00:10:50.430 --> 00:10:52.850
It's only the output
that's a vector.
00:10:52.850 --> 00:10:55.630
So let's go back and,
just as I say, arrowize
00:10:55.630 --> 00:10:59.230
or vectorize whatever has
to be vectorized here.
00:10:59.230 --> 00:11:01.990
So I now have a new
definition, intuitively.
00:11:01.990 --> 00:11:03.650
The limit of the
vector function f
00:11:03.650 --> 00:11:08.680
of x as x approaches a is the
vector L means that the vector
00:11:08.680 --> 00:11:12.725
f of x is near the vector L,
provided that the scalar x
00:11:12.725 --> 00:11:14.720
is near the scalar a.
00:11:14.720 --> 00:11:18.220
Now the point is that
the "x is near a"
00:11:18.220 --> 00:11:20.770
doesn't need any further
interpretation because a
00:11:20.770 --> 00:11:22.510
and x are still scalars.
00:11:22.510 --> 00:11:24.770
The question that comes
up now is twofold.
00:11:24.770 --> 00:11:29.070
First of all, does it make sense
to say that f of x is near L?
00:11:29.070 --> 00:11:31.950
Does that even make sense when
you're talking about vectors?
00:11:31.950 --> 00:11:35.120
The second thing is,
if it does make sense,
00:11:35.120 --> 00:11:37.110
is it the meaning that
we want it to have?
00:11:37.110 --> 00:11:39.970
Does it capture our
intuitive feeling?
00:11:39.970 --> 00:11:43.550
Let's take the questions
in order of appearance.
00:11:43.550 --> 00:11:47.300
First of all, what does it mean
to say that f of x-- the vector
00:11:47.300 --> 00:11:49.690
f of x-- is near the vector L?
00:11:49.690 --> 00:11:51.970
In fact, let's generalize that.
00:11:51.970 --> 00:11:54.210
Given any two
vectors A and B, what
00:11:54.210 --> 00:11:59.600
does it mean to say that the
vector A is near the vector B?
00:11:59.600 --> 00:12:01.270
Now what my claim is, is this.
00:12:01.270 --> 00:12:04.504
Let's draw the vectors A and
B as arrows in the plane here.
00:12:04.504 --> 00:12:06.920
Let's assume that they start
at a common origin, which you
00:12:06.920 --> 00:12:08.850
can always assume, of course.
00:12:08.850 --> 00:12:13.640
To say that A is nearly equal
to B somehow means what?
00:12:13.640 --> 00:12:17.520
That the vector A should be very
nearly equal to the vector B.
00:12:17.520 --> 00:12:20.060
Well remember, if
two vectors start
00:12:20.060 --> 00:12:22.120
at a common point,
the only way that they
00:12:22.120 --> 00:12:25.410
can be equal is if
their heads coincide.
00:12:25.410 --> 00:12:28.820
Consequently, with A and B
starting at a common point,
00:12:28.820 --> 00:12:31.840
to say that A is
nearly equal to B
00:12:31.840 --> 00:12:34.340
should mean that the distance
between the head of A
00:12:34.340 --> 00:12:36.890
and the head of B
should be small.
00:12:36.890 --> 00:12:39.530
But how do we state the
distance between the head of A
00:12:39.530 --> 00:12:40.570
and the head of B?
00:12:40.570 --> 00:12:44.130
Notice that that has a very
convenient numerical form.
00:12:44.130 --> 00:12:47.700
Namely, the vector
that joins A to B
00:12:47.700 --> 00:12:50.400
can be written as either
A minus B or B minus A,
00:12:50.400 --> 00:12:52.390
depending on what
sense you put on this.
00:12:52.390 --> 00:12:53.720
We don't care about the sense.
00:12:53.720 --> 00:12:55.650
All we care about
is the magnitude.
00:12:55.650 --> 00:13:00.000
Let's just call this length
the magnitude of A minus B.
00:13:00.000 --> 00:13:01.230
And now we're in business.
00:13:01.230 --> 00:13:03.190
Namely, we say, lookit.
00:13:03.190 --> 00:13:07.800
To say that A is near B means
that A is nearly equal to B,
00:13:07.800 --> 00:13:13.190
which in turn means that
the magnitude of A minus B--
00:13:13.190 --> 00:13:16.110
the magnitude of the
vector-- what vector?--
00:13:16.110 --> 00:13:19.290
the vector A minus the
vector B-- is small.
00:13:19.290 --> 00:13:22.110
And keep in mind that
I'm now using 'small'
00:13:22.110 --> 00:13:23.840
as I did arithmetically.
00:13:23.840 --> 00:13:26.530
Because even though
A and B are vectors,
00:13:26.530 --> 00:13:31.000
and so also is A minus B,
the magnitude of a vector
00:13:31.000 --> 00:13:32.320
is a number.
00:13:32.320 --> 00:13:34.940
Note this very important thing.
00:13:34.940 --> 00:13:36.580
Even when I'm
dealing with vectors,
00:13:36.580 --> 00:13:40.580
as soon as I mention magnitude,
I'm talking about a number.
00:13:40.580 --> 00:13:42.690
In other words
then, I now define
00:13:42.690 --> 00:13:47.320
A to be near B to mean that
the magnitude of the difference
00:13:47.320 --> 00:13:51.690
between A and B is small.
00:13:51.690 --> 00:13:54.400
And again, question
two-- does that
00:13:54.400 --> 00:13:56.320
caption my intuitive feeling?
00:13:56.320 --> 00:13:57.710
And the answer is, yes, it does.
00:13:57.710 --> 00:14:03.500
I want A to be near B to mean
that they nearly coincide.
00:14:03.500 --> 00:14:06.490
And that is the same as saying
that the magnitude of A minus B
00:14:06.490 --> 00:14:07.380
is small.
00:14:07.380 --> 00:14:09.730
At any rate, having
gone through this
00:14:09.730 --> 00:14:11.790
from a fairly intuitive
point of view,
00:14:11.790 --> 00:14:15.720
let's now revisit limits
more rigorously and rewrite
00:14:15.720 --> 00:14:19.414
our limit definition in
terms of epsilons and deltas.
00:14:19.414 --> 00:14:21.830
And by the way, I hope this
doesn't look like anything new
00:14:21.830 --> 00:14:24.060
to you, what I'm going
to be talking about next.
00:14:24.060 --> 00:14:26.380
It's our old definition
of limit that
00:14:26.380 --> 00:14:29.540
was the backbone of the entire
part one of this course.
00:14:29.540 --> 00:14:31.960
I not only hope that it
looks familiar to you,
00:14:31.960 --> 00:14:35.800
I hope that you read this
thing almost subconsciously
00:14:35.800 --> 00:14:37.520
as second nature by now.
00:14:37.520 --> 00:14:38.620
But the statement is what?
00:14:38.620 --> 00:14:42.120
The limit of f of x as x
approaches a equals L means,
00:14:42.120 --> 00:14:44.430
given any epsilon
greater than 0,
00:14:44.430 --> 00:14:47.520
we can find a delta
greater than 0 such
00:14:47.520 --> 00:14:52.040
that whenever the absolute value
of x minus a is less than delta
00:14:52.040 --> 00:14:54.340
but greater than
0, the implication
00:14:54.340 --> 00:14:57.670
is that the absolute
value of f of x minus L
00:14:57.670 --> 00:14:59.510
is less than epsilon.
00:14:59.510 --> 00:15:02.730
Now if I go back to my
structural properties again,
00:15:02.730 --> 00:15:06.340
it seems to me that what my
first approximation, at least--
00:15:06.340 --> 00:15:10.150
for a rigorous definition to
limits of vector functions--
00:15:10.150 --> 00:15:12.330
should be to just,
as I did before,
00:15:12.330 --> 00:15:14.930
read through this
definition verbatim.
00:15:14.930 --> 00:15:17.890
Don't change a thing,
but just vectorize
00:15:17.890 --> 00:15:19.970
what has to be vectorized.
00:15:19.970 --> 00:15:22.740
And because this definition
that I'm going to give
00:15:22.740 --> 00:15:26.000
is so important, I
elect to rewrite it.
00:15:26.000 --> 00:15:27.740
And what you're
going to see next
00:15:27.740 --> 00:15:30.390
is nothing more
than a carbon copy,
00:15:30.390 --> 00:15:34.570
so to speak, of this, only
with appropriate arrows being
00:15:34.570 --> 00:15:35.490
put in.
00:15:35.490 --> 00:15:39.150
Namely, I am going to give,
as my rigorous definition
00:15:39.150 --> 00:15:41.350
of the limit of f
of x as x approaches
00:15:41.350 --> 00:15:44.920
a equals the vector L--
it's going to mean what?
00:15:44.920 --> 00:15:48.950
Given epsilon greater than
0, I can find delta greater
00:15:48.950 --> 00:15:53.240
than 0 such that whenever the
absolute value of x minus a
00:15:53.240 --> 00:15:57.340
is less than delta but greater
than 0, the implication is
00:15:57.340 --> 00:16:00.175
that the magnitude-- see I
don't say absolute value now,
00:16:00.175 --> 00:16:02.050
because notice, I'm not
dealing with numbers.
00:16:02.050 --> 00:16:06.390
These are vectors-- but the
magnitude of f of x minus L
00:16:06.390 --> 00:16:09.366
must be less than epsilon.
00:16:09.366 --> 00:16:13.260
Now you see what I've done: I've
obtained the second definition
00:16:13.260 --> 00:16:16.830
from the first simply by
appropriate vectorization.
00:16:16.830 --> 00:16:20.870
What I have to make
sure of is what?
00:16:20.870 --> 00:16:22.760
That this thing
still makes sense.
00:16:22.760 --> 00:16:24.380
And notice that it does.
00:16:24.380 --> 00:16:26.950
Notice that what this says,
in plain English is, what?
00:16:26.950 --> 00:16:34.200
That if x is sufficiently
close to a-- what?
00:16:34.200 --> 00:16:36.920
The magnitude of the difference
between these two vectors
00:16:36.920 --> 00:16:38.410
is as small as we wish.
00:16:38.410 --> 00:16:41.289
But we call that magnitude what?
00:16:41.289 --> 00:16:42.830
The distance between
the two vectors.
00:16:42.830 --> 00:16:45.020
This says what?
00:16:45.020 --> 00:16:49.160
Correlating our formal language
with our informal language,
00:16:49.160 --> 00:16:50.640
this simply says what?
00:16:50.640 --> 00:16:54.170
That we can make f
of x as near to L
00:16:54.170 --> 00:16:58.640
as we wish just by picking
x sufficiently close to a.
00:16:58.640 --> 00:17:03.240
So now what that tells us is
that the new limit definition
00:17:03.240 --> 00:17:06.626
makes sense, just by
mimicking the old definition.
00:17:06.626 --> 00:17:08.000
And what do we
mean by mimicking?
00:17:08.000 --> 00:17:11.180
We mean vectorize
the old definition,
00:17:11.180 --> 00:17:13.730
go through it word for
word and put in vectors
00:17:13.730 --> 00:17:15.579
in appropriate places.
00:17:15.579 --> 00:17:17.670
Now let's see what this
means structurally.
00:17:17.670 --> 00:17:21.200
That's the whole key to our
particular lecture today:
00:17:21.200 --> 00:17:23.109
the structural value of this.
00:17:23.109 --> 00:17:26.380
For example, what were
some of the building blocks
00:17:26.380 --> 00:17:29.860
that we based calculus
of a single variable on?
00:17:29.860 --> 00:17:33.160
Remember, derivative
was defined as a limit,
00:17:33.160 --> 00:17:35.890
and therefore all of our
properties about derivatives
00:17:35.890 --> 00:17:37.660
followed from limit properties.
00:17:37.660 --> 00:17:42.830
Well, among the limit properties
that we had for part one where,
00:17:42.830 --> 00:17:43.330
what?
00:17:43.330 --> 00:17:45.220
Input and output were scalars.
00:17:45.220 --> 00:17:49.810
The limit of the sum was equal
to the sum of the limits.
00:17:49.810 --> 00:17:52.480
And the limit of a product
is equal to the product
00:17:52.480 --> 00:17:53.610
the limits.
00:17:53.610 --> 00:17:55.690
And remember, in
all of our proofs,
00:17:55.690 --> 00:17:58.900
we use these particular
structural properties.
00:17:58.900 --> 00:17:59.400
You see?
00:17:59.400 --> 00:18:01.020
The next thing we
would like to know--
00:18:01.020 --> 00:18:03.300
see after all, these
structural properties,
00:18:03.300 --> 00:18:04.800
even though they're
called theorems,
00:18:04.800 --> 00:18:07.290
essentially, they become
the rules of the game
00:18:07.290 --> 00:18:08.560
once we've proven them.
00:18:08.560 --> 00:18:10.430
In other words,
once we've proved
00:18:10.430 --> 00:18:14.050
using epsilons and deltas
that this happens to be true,
00:18:14.050 --> 00:18:18.030
notice that we never again
use the fact that these are
00:18:18.030 --> 00:18:22.100
epsilons and deltas, that we
have epsilons and deltas when
00:18:22.100 --> 00:18:24.870
we use this particular
result. We just write it down.
00:18:24.870 --> 00:18:27.730
Because once we've proven
it, we can always use it.
00:18:27.730 --> 00:18:29.720
In other words, what
I'm really saying is,
00:18:29.720 --> 00:18:31.440
let me vectorize this.
00:18:35.392 --> 00:18:37.640
And I'll put a question
mark now, because you see,
00:18:37.640 --> 00:18:39.450
I don't know if this
is true for vectors.
00:18:39.450 --> 00:18:40.867
I don't know if
the limit of a sum
00:18:40.867 --> 00:18:42.325
is the sum of the
limits when we're
00:18:42.325 --> 00:18:44.360
dealing with vectors
rather than with scalars.
00:18:44.360 --> 00:18:45.980
But notice what I've done.
00:18:45.980 --> 00:18:51.020
I have structurally plagiarized
from my original definition.
00:18:51.020 --> 00:18:53.069
All I did was I vectorized it.
00:18:53.069 --> 00:18:54.610
You see, what I'm
going to have to do
00:18:54.610 --> 00:18:57.050
is to check from my
original definition
00:18:57.050 --> 00:18:59.562
whether these properties
still hold true.
00:18:59.562 --> 00:19:01.270
I'm going to talk
about that in a minute.
00:19:01.270 --> 00:19:03.380
Let me just continue
on for a while.
00:19:03.380 --> 00:19:05.430
See, similarly over
here, we talked
00:19:05.430 --> 00:19:07.350
about the limit of
a product equaling
00:19:07.350 --> 00:19:08.780
the product of the limits.
00:19:08.780 --> 00:19:10.840
So again, we'd like to
use results like that.
00:19:10.840 --> 00:19:12.590
And somebody says,
well, let's vectorize.
00:19:14.684 --> 00:19:16.350
I want to show you
what I mean by saying
00:19:16.350 --> 00:19:19.389
that after you vectorize,
you have to be darn careful
00:19:19.389 --> 00:19:21.430
that you don't just go
around saying, look at it,
00:19:21.430 --> 00:19:22.090
this is legal.
00:19:22.090 --> 00:19:24.080
I put arrows over everything.
00:19:24.080 --> 00:19:26.410
Sure it's legal,
but you may not have
00:19:26.410 --> 00:19:28.240
anything that's worth keeping.
00:19:28.240 --> 00:19:29.820
Look at this expression.
00:19:29.820 --> 00:19:33.030
What are f and g in this case?
00:19:33.030 --> 00:19:36.090
The way I've written it,
these happen to be vectors.
00:19:36.090 --> 00:19:37.280
See, they're both arrowized.
00:19:37.280 --> 00:19:37.780
Right?
00:19:37.780 --> 00:19:38.870
They're vectors.
00:19:38.870 --> 00:19:41.110
Have we talked about
the ordinary product
00:19:41.110 --> 00:19:42.650
of two vector functions?
00:19:42.650 --> 00:19:43.860
And the answer is no.
00:19:43.860 --> 00:19:45.930
When we have two
vector functions,
00:19:45.930 --> 00:19:48.710
we must either have
a dot or a cross
00:19:48.710 --> 00:19:51.210
in here-- the dot product
or the cross product.
00:19:51.210 --> 00:19:53.210
And if we're not going
to put a dot in here,
00:19:53.210 --> 00:19:57.190
the best we can talk about
is scalar multiplication.
00:19:57.190 --> 00:20:00.500
In other words, let me-- before
you tend to memorize this,
00:20:00.500 --> 00:20:02.370
let me cross that out.
00:20:02.370 --> 00:20:04.350
Because this is ambiguous.
00:20:04.350 --> 00:20:05.610
It doesn't make sense.
00:20:05.610 --> 00:20:07.610
And what I am saying is
that unfortunately-- not
00:20:07.610 --> 00:20:09.943
unfortunately, but if I want
to be rigorous about this--
00:20:09.943 --> 00:20:12.430
there happens to be three
interpretations here.
00:20:12.430 --> 00:20:14.190
Namely, I can do what?
00:20:14.190 --> 00:20:16.790
I can think of f(x)
as being a scalar
00:20:16.790 --> 00:20:19.180
and g(x) as being a vector.
00:20:19.180 --> 00:20:24.190
Or I can think of f(x) and g(x)
as both being vector functions,
00:20:24.190 --> 00:20:26.360
but in one case
having the dot product
00:20:26.360 --> 00:20:28.180
and in the other case,
the cross product.
00:20:28.180 --> 00:20:32.120
In other words, to vectorize
this particular equation,
00:20:32.120 --> 00:20:37.800
I guess what I have
to do is write down
00:20:37.800 --> 00:20:39.630
these three possibilities.
00:20:39.630 --> 00:20:41.950
Namely, notice in this
case, this is what?
00:20:41.950 --> 00:20:45.360
This is a scalar multiple
of a vector function.
00:20:45.360 --> 00:20:48.270
Granted that the scalar multiple
is a variable in this case,
00:20:48.270 --> 00:20:49.300
this still makes sense.
00:20:49.300 --> 00:20:54.170
Namely, for a fixed x-- for a
fixed x, f of x is a constant,
00:20:54.170 --> 00:20:55.830
and g of x is a vector.
00:20:55.830 --> 00:20:59.630
A constant times a
vector is a vector.
00:20:59.630 --> 00:21:04.390
For a fixed x, f and g
here are fixed vectors.
00:21:04.390 --> 00:21:07.320
And I can talk about the
dot product of two vectors.
00:21:07.320 --> 00:21:11.050
And for a fixed x over
here, f of x and g of x
00:21:11.050 --> 00:21:13.850
are again fixed vectors,
and consequently, I
00:21:13.850 --> 00:21:15.630
can talk about
their cross product.
00:21:15.630 --> 00:21:17.900
And the question is, will
these properties still
00:21:17.900 --> 00:21:20.930
be true when we're dealing
with vector functions?
00:21:20.930 --> 00:21:22.920
The answer happens to be yes.
00:21:22.920 --> 00:21:25.260
But it's not yes automatically.
00:21:25.260 --> 00:21:28.970
In other words, what we're
going to do in the notes is
00:21:28.970 --> 00:21:31.770
to mimic-- to prove these
results for vectors, what we
00:21:31.770 --> 00:21:35.610
are going to do is to
mimic every single proof
00:21:35.610 --> 00:21:38.020
that we gave in the
scalar case, only
00:21:38.020 --> 00:21:41.450
replacing scalars by
appropriate vectors
00:21:41.450 --> 00:21:43.720
whenever this is
supposed to happen.
00:21:43.720 --> 00:21:46.180
The trouble is that
certain results which
00:21:46.180 --> 00:21:48.880
were true for scalars
may not automatically
00:21:48.880 --> 00:21:50.710
be true for vectors.
00:21:50.710 --> 00:21:52.520
Let me give you an example.
00:21:52.520 --> 00:21:54.780
Somehow or other to prove
that the limit of a sum
00:21:54.780 --> 00:21:56.400
was equal to the
sum of the limits,
00:21:56.400 --> 00:21:59.470
we used the property
of absolute values
00:21:59.470 --> 00:22:02.600
that said that the absolute
value of a sum was less than
00:22:02.600 --> 00:22:05.290
or equal to the sum of
the absolute values.
00:22:05.290 --> 00:22:07.130
And we proved that for numbers.
00:22:07.130 --> 00:22:09.150
Suppose I now vectorize this.
00:22:14.750 --> 00:22:15.480
See?
00:22:15.480 --> 00:22:17.530
This now means
magnitudes, right?
00:22:17.530 --> 00:22:20.310
And if a and b are any
vectors in the plane,
00:22:20.310 --> 00:22:23.090
a and b no longer have to
be in the same direction--
00:22:23.090 --> 00:22:27.390
how do we know that vector
addition has the same magnitude
00:22:27.390 --> 00:22:30.240
property that
scalar addition has?
00:22:30.240 --> 00:22:33.010
See, can we be sure that
this recipe is true?
00:22:33.010 --> 00:22:36.510
Well let's go and see
what this recipe means.
00:22:36.510 --> 00:22:39.400
By the way, this does happen to
be true in vector arithmetic.
00:22:39.400 --> 00:22:42.130
And it happens to be known
as the triangle inequality.
00:22:42.130 --> 00:22:44.870
And the reason for that is,
if I draw a triangle calling
00:22:44.870 --> 00:22:47.700
two of the sides a
and b, respectively,
00:22:47.700 --> 00:22:51.830
and the third side c, we do
know from elementary geometry
00:22:51.830 --> 00:22:56.030
that the sum of the lengths
of two sides of a triangle
00:22:56.030 --> 00:22:58.900
is greater than or equal to
the length of the third side
00:22:58.900 --> 00:23:00.040
of the triangle.
00:23:00.040 --> 00:23:03.400
What I'm driving at now is-- let
me just vectorize this and show
00:23:03.400 --> 00:23:05.480
you something that I
think is very cute here.
00:23:05.480 --> 00:23:08.910
If I put arrows here, this
becomes the vector a, now;
00:23:08.910 --> 00:23:12.896
this becomes the vector b;
this becomes the vector c.
00:23:12.896 --> 00:23:13.750
OK?
00:23:13.750 --> 00:23:16.590
But in terms of our
definition of vector addition,
00:23:16.590 --> 00:23:20.040
noticed that c goes from the
tail of a to the head of b.
00:23:20.040 --> 00:23:23.380
a and b are lined up
properly for addition.
00:23:23.380 --> 00:23:27.400
So this is the vector a plus b.
00:23:27.400 --> 00:23:29.940
Now state the
triangle inequality
00:23:29.940 --> 00:23:31.620
in terms of this picture.
00:23:31.620 --> 00:23:34.280
The triangle
inequality says what?
00:23:34.280 --> 00:23:37.330
The third side of the triangle--
well what length is this?
00:23:37.330 --> 00:23:39.630
If the vector is a
plus b, the length
00:23:39.630 --> 00:23:43.540
is the magnitude of a
plus b-- must be less than
00:23:43.540 --> 00:23:46.720
or equal to the sum of the
lengths of the other two sides.
00:23:46.720 --> 00:23:52.730
But those are just this.
00:23:52.730 --> 00:23:55.960
And that verifies the
recipe that we want.
00:23:55.960 --> 00:23:57.910
In other words, it's
going to turn out
00:23:57.910 --> 00:24:03.060
that every vector property
that we need-- what
00:24:03.060 --> 00:24:05.690
do we mean by need-- that
happened to be true for scalar
00:24:05.690 --> 00:24:09.220
calculus-- is going to be
sufficient in vector calculus
00:24:09.220 --> 00:24:09.910
as well.
00:24:09.910 --> 00:24:13.570
For example, when we proved
the product rule for scalars--
00:24:13.570 --> 00:24:15.570
not the product rule, but
the limit of a product
00:24:15.570 --> 00:24:17.780
was the product of the
limits-- we used such things
00:24:17.780 --> 00:24:20.770
as the absolute
value of a product
00:24:20.770 --> 00:24:23.420
was equal to the product
of the absolute values.
00:24:23.420 --> 00:24:26.330
Notice, for example, in
terms of case one over here,
00:24:26.330 --> 00:24:31.080
if I vectorize b and leave
a as a scalar, the magnitude
00:24:31.080 --> 00:24:35.330
of a times the vector b
is equal to the magnitude
00:24:35.330 --> 00:24:38.730
of a times the magnitude of b,
just by the definition of what
00:24:38.730 --> 00:24:40.350
we meant by a scalar multiple.
00:24:40.350 --> 00:24:42.600
After all, a times b meant what?
00:24:42.600 --> 00:24:45.670
You just kept the
direction of b constant,
00:24:45.670 --> 00:24:50.840
but multiplied b by
the magnitude of a.
00:24:50.840 --> 00:24:51.540
Meaning what?
00:24:51.540 --> 00:24:54.250
By a, if was a positive;
minus a, if a was negative,
00:24:54.250 --> 00:24:55.975
this is certainly true.
00:24:55.975 --> 00:24:57.850
By the way, there is a
very important caution
00:24:57.850 --> 00:24:59.470
that I'd like to warn you about.
00:24:59.470 --> 00:25:01.760
And this is done in great
detail in the notes.
00:25:01.760 --> 00:25:06.650
It's very important to point
out that when I vectorize this
00:25:06.650 --> 00:25:08.820
in terms of a dot
and a cross product,
00:25:08.820 --> 00:25:12.510
it is not true that the
magnitude of a dot b
00:25:12.510 --> 00:25:16.950
is equal to the magnitude of
a times the magnitude of b.
00:25:16.950 --> 00:25:18.450
You see, what I'm
really saying here
00:25:18.450 --> 00:25:25.320
is, how was a dot b defined? a
dot b, by definition, was what?
00:25:25.320 --> 00:25:29.110
It was the magnitude of a
times the magnitude of b,
00:25:29.110 --> 00:25:32.540
times the cosine of the
angle between a and b.
00:25:32.540 --> 00:25:36.709
Notice that if I take
magnitudes here--
00:25:36.709 --> 00:25:39.250
I want to keep this separate,
so you can see what I'm talking
00:25:39.250 --> 00:25:41.300
about here--
00:25:41.300 --> 00:25:41.800
Lookit.
00:25:41.800 --> 00:25:44.326
This factor here can
be no bigger than 1.
00:25:44.326 --> 00:25:46.200
But in general, it's
going to be less than 1.
00:25:46.200 --> 00:25:48.950
In other words, notice that
the magnitude of a dot b
00:25:48.950 --> 00:25:52.410
is actually less than or equal
to the magnitude of a times
00:25:52.410 --> 00:25:53.520
the magnitude of b.
00:25:53.520 --> 00:25:57.800
In fact, its equality holds
only if this is a 0 degree
00:25:57.800 --> 00:25:59.960
angle or 180 degree angle.
00:25:59.960 --> 00:26:02.320
Because only then is the
magnitude of the cosine
00:26:02.320 --> 00:26:03.560
equal to 1.
00:26:03.560 --> 00:26:06.057
Similar result holds
for the cross product.
00:26:06.057 --> 00:26:07.640
The interesting point
is-- and I'm not
00:26:07.640 --> 00:26:09.181
going to take the
time to do it here,
00:26:09.181 --> 00:26:12.050
because I want this lecture
to be basically an overview--
00:26:12.050 --> 00:26:16.190
but the important point is
that you don't need equality
00:26:16.190 --> 00:26:17.370
to prove our limit theorems.
00:26:17.370 --> 00:26:19.190
Remember, every one
of our limit theorems
00:26:19.190 --> 00:26:21.470
was a string of inequalities.
00:26:21.470 --> 00:26:23.080
And actually-- and
as I say, I'm going
00:26:23.080 --> 00:26:24.980
to do this in more
detail in the notes--
00:26:24.980 --> 00:26:28.460
the only result that we
needed to prove theorems
00:26:28.460 --> 00:26:32.190
even in the part one
section of our course
00:26:32.190 --> 00:26:36.100
was the fact that the less
than or equal part be valid.
00:26:36.100 --> 00:26:38.830
The fact that it was equal when
we were dealing with numbers
00:26:38.830 --> 00:26:40.450
was like frosting on the cake.
00:26:40.450 --> 00:26:43.910
We didn't need that
strong a condition.
00:26:43.910 --> 00:26:46.190
The key overview that
I want you to get
00:26:46.190 --> 00:26:48.320
from this present
discussion is that, when
00:26:48.320 --> 00:26:51.750
we vectorize our
definition of limit,
00:26:51.750 --> 00:26:55.680
that that new definition,
having the same structure
00:26:55.680 --> 00:26:58.840
as the old-- meaning that all
the properties of magnitudes
00:26:58.840 --> 00:27:01.590
of vectors that we
need are carried over
00:27:01.590 --> 00:27:04.870
from absolute values of
numbers, all previous limit
00:27:04.870 --> 00:27:07.550
theorems-- what do I mean by
"all previous limit theorems"?
00:27:07.550 --> 00:27:11.080
I mean all the limit theorems
of part one of this course
00:27:11.080 --> 00:27:13.330
are still valid.
00:27:13.330 --> 00:27:16.300
And with that in mind,
I can now proceed
00:27:16.300 --> 00:27:18.600
to differential calculus.
00:27:18.600 --> 00:27:19.559
Namely what?
00:27:19.559 --> 00:27:21.100
I'm going to do the
same thing again.
00:27:21.100 --> 00:27:24.410
I write down the definition
of derivative as it
00:27:24.410 --> 00:27:26.710
was in part one of our course.
00:27:26.710 --> 00:27:31.440
And now what I do is I just
vectorize everything in sight,
00:27:31.440 --> 00:27:33.390
provided it's supposed
to be vectorized.
00:27:33.390 --> 00:27:36.370
Remember x and delta
x are scalars here.
00:27:36.370 --> 00:27:38.590
f is the function.
00:27:38.590 --> 00:27:40.720
I just vectorize our definition.
00:27:40.720 --> 00:27:42.870
The first thing I
have to do is to check
00:27:42.870 --> 00:27:45.350
to see whether the new
expression makes sense.
00:27:45.350 --> 00:27:47.970
Notice that the numerator
of my bracketed expression
00:27:47.970 --> 00:27:49.240
is now a vector.
00:27:49.240 --> 00:27:51.180
The denominator is a scalar.
00:27:51.180 --> 00:27:54.520
And a vector divided by a scalar
makes perfectly good sense.
00:27:54.520 --> 00:27:57.590
In other words, this
can be read as what?
00:27:57.590 --> 00:28:01.605
The scalar multiple 1 over
delta x times the vector
00:28:01.605 --> 00:28:04.930
f of x plus delta
x minus f of x.
00:28:04.930 --> 00:28:07.570
So this definition
makes very good sense.
00:28:07.570 --> 00:28:10.676
It captures the meaning
of average rate of change
00:28:10.676 --> 00:28:11.800
because you see, it's what?
00:28:11.800 --> 00:28:14.950
It's the total change
in f over the change
00:28:14.950 --> 00:28:16.720
in x equal to delta x.
00:28:16.720 --> 00:28:18.430
So it's an average
rate of change.
00:28:18.430 --> 00:28:21.000
Again, that's discussed
in more detail in the text
00:28:21.000 --> 00:28:23.030
and in our notes,
and in the exercises.
00:28:23.030 --> 00:28:26.460
But the point is that since
every derivative property
00:28:26.460 --> 00:28:28.980
that we had in part
one of our course
00:28:28.980 --> 00:28:31.700
followed from our
limit properties,
00:28:31.700 --> 00:28:33.540
the fact that the
limit properties are
00:28:33.540 --> 00:28:37.440
the same for vectors as
they are for scalars now
00:28:37.440 --> 00:28:41.850
means that all derivative
formulas are still valid.
00:28:41.850 --> 00:28:45.157
In other words, the product
rule is still going to hold.
00:28:45.157 --> 00:28:46.740
The derivative of a
sum is still going
00:28:46.740 --> 00:28:48.410
to be the sum of
the derivatives.
00:28:48.410 --> 00:28:51.010
The derivative of a constant
is still going to be 0.
00:28:51.010 --> 00:28:52.060
And keep this in mind.
00:28:52.060 --> 00:28:53.200
It's very important.
00:28:53.200 --> 00:28:56.460
I could re-derive every
single one of these results
00:28:56.460 --> 00:28:59.650
from scratch as if
scalar calculus had never
00:28:59.650 --> 00:29:04.820
been invented, just by using my
basic epsilon-delta definition.
00:29:04.820 --> 00:29:08.190
The beauty of structure is, is
that since the structures are
00:29:08.190 --> 00:29:10.570
alike-- you see,
since they are played
00:29:10.570 --> 00:29:15.200
by the same rules of the game--
the theorems of vector calculus
00:29:15.200 --> 00:29:18.545
are going to be precisely
those of scalar calculus.
00:29:18.545 --> 00:29:22.370
And I don't have to take the
time to re-derive them all.
00:29:22.370 --> 00:29:24.740
As I say in the notes,
I re-derive a few
00:29:24.740 --> 00:29:27.690
just so that you can get an
idea of how this transliteration
00:29:27.690 --> 00:29:28.760
takes place.
00:29:28.760 --> 00:29:32.480
At any rate, let me close
today's lesson with an example.
00:29:32.480 --> 00:29:35.290
Let's take motion in a plane.
00:29:35.290 --> 00:29:38.810
Let's suppose we have a
particle moving along a curve C.
00:29:38.810 --> 00:29:40.540
And that the motion
of the particle
00:29:40.540 --> 00:29:43.930
is given in parametric form,
meaning we know both the x-
00:29:43.930 --> 00:29:47.870
and the y-coordinates of
the particle at any time t,
00:29:47.870 --> 00:29:49.890
as functions of t.
00:29:49.890 --> 00:29:53.110
Notice, by the way,
that what requires
00:29:53.110 --> 00:29:57.620
two equations in scalar form can
be written as a single vector
00:29:57.620 --> 00:30:01.880
equation, namely, at time
t-- let's say at time t,
00:30:01.880 --> 00:30:03.790
the particle was over here.
00:30:03.790 --> 00:30:08.440
Notice that if we now let r
be the vector from the origin
00:30:08.440 --> 00:30:13.200
to the position of the particle
that r is a vector, right?
00:30:13.200 --> 00:30:16.530
Because to specify r, I not
only have to give its length,
00:30:16.530 --> 00:30:17.970
I have to give its direction.
00:30:17.970 --> 00:30:22.601
r is a vector which
varies with our scalar t.
00:30:22.601 --> 00:30:23.100
All right?
00:30:23.100 --> 00:30:26.400
So r is a vector
function of the scalar t.
00:30:26.400 --> 00:30:29.880
Now what is the
i component of r?
00:30:29.880 --> 00:30:32.710
Well, the i component is x.
00:30:32.710 --> 00:30:34.920
And the j component is y.
00:30:34.920 --> 00:30:40.160
Notice that the pair of
simultaneous parametric scalar
00:30:40.160 --> 00:30:43.820
equations x equals x
of t, y equals y of t
00:30:43.820 --> 00:30:48.230
can be written as the single
vector equation r of t
00:30:48.230 --> 00:30:54.040
equals x of t i plus y of t
j, where x of t and y of t
00:30:54.040 --> 00:30:57.580
are scalar functions of
the scalar variable t.
00:30:57.580 --> 00:30:59.690
OK?
00:30:59.690 --> 00:31:01.820
Now the question
comes up, wouldn't it
00:31:01.820 --> 00:31:04.380
be nice to just take dr/dt here?
00:31:04.380 --> 00:31:08.620
Well, I mean, that's about
as motivated as you can get.
00:31:08.620 --> 00:31:09.180
Meaning what?
00:31:09.180 --> 00:31:10.138
Let's see what happens.
00:31:10.138 --> 00:31:11.550
I know how to differentiate now.
00:31:11.550 --> 00:31:14.500
I'm going to use the fact that
the derivative of a product
00:31:14.500 --> 00:31:16.530
of a sum is a sum
of the derivatives,
00:31:16.530 --> 00:31:18.290
that i and j are constants.
00:31:18.290 --> 00:31:21.310
And a constant times a
function, to differentiate that,
00:31:21.310 --> 00:31:24.260
you skip over the constant and
differentiate the function.
00:31:24.260 --> 00:31:27.320
x of t and y of t
are scalar functions,
00:31:27.320 --> 00:31:30.050
and I already know how to
differentiate scalar functions.
00:31:30.050 --> 00:31:34.030
So all I'm going to assume
now is that x of t and y of t
00:31:34.030 --> 00:31:37.420
are differentiable scalar
functions, which means now
00:31:37.420 --> 00:31:39.830
that instead of just talking
about motion in a plane,
00:31:39.830 --> 00:31:41.980
I'm assuming that
the motion is smooth.
00:31:41.980 --> 00:31:45.380
At any rate, I now
differentiate term by term.
00:31:45.380 --> 00:31:46.769
And look what I get.
00:31:46.769 --> 00:31:47.310
This is what?
00:31:47.310 --> 00:31:51.730
It's dx/dt times i
plus dy/dt times j.
00:31:51.730 --> 00:31:55.120
In other words, the dr/dt
is this particular vector.
00:31:55.120 --> 00:31:57.850
And this particular
vector is fascinating.
00:31:57.850 --> 00:31:59.380
Why is it fascinating?
00:31:59.380 --> 00:32:02.700
Well, for one thing, let's
compute its magnitude.
00:32:02.700 --> 00:32:06.300
The magnitude of any vector in
i and j components is, what?
00:32:06.300 --> 00:32:09.510
The square root of the sum of
the squares of the components.
00:32:09.510 --> 00:32:11.010
In this case, that's
the square root
00:32:11.010 --> 00:32:16.350
of dx/dt squared
plus dy/dt squared.
00:32:16.350 --> 00:32:18.360
But remember from part
one of our course,
00:32:18.360 --> 00:32:23.250
this is exactly the magnitude
of ds/dt where s is arc length.
00:32:23.250 --> 00:32:25.900
Remember I put the
absolute value sign in here
00:32:25.900 --> 00:32:29.250
because we always take
the positive square root.
00:32:29.250 --> 00:32:30.670
We're assuming
that arc length is
00:32:30.670 --> 00:32:35.290
traversed in a given direction
at a particular time t.
00:32:35.290 --> 00:32:36.920
But what is ds/dt?
00:32:36.920 --> 00:32:40.020
It's the change in arc
length with respect to time.
00:32:40.020 --> 00:32:43.860
And that's precisely what we
mean by speed along the curve.
00:32:43.860 --> 00:32:48.210
On the other hand, if we look at
the slope of dr/dt, it's what?
00:32:48.210 --> 00:32:50.230
It's the slope-- it's
determined by what?
00:32:50.230 --> 00:32:53.180
You take the j component,
which is dy/dt,
00:32:53.180 --> 00:32:56.480
divided by the i
component, which is dx/dt.
00:32:56.480 --> 00:32:59.610
By the chain rule, we
know that that's dy/dx.
00:32:59.610 --> 00:33:03.400
Therefore, the vector
dr/dt has its magnitude
00:33:03.400 --> 00:33:05.640
equal to the speed
along the curve.
00:33:05.640 --> 00:33:08.480
And its direction is
tangential to the curve.
00:33:08.480 --> 00:33:10.800
And what better
motivation than that
00:33:10.800 --> 00:33:13.280
to define a velocity vector?
00:33:13.280 --> 00:33:15.670
After all, we've already done
that in elementary physics.
00:33:15.670 --> 00:33:18.130
In elementary physics, what
was the velocity vector
00:33:18.130 --> 00:33:19.320
defined to be?
00:33:19.320 --> 00:33:21.440
At a given point,
it was the vector
00:33:21.440 --> 00:33:24.040
whose direction to the
curve-- whose direction
00:33:24.040 --> 00:33:26.670
was tangential to the
path at that point
00:33:26.670 --> 00:33:29.120
and whose magnitude
was numerically
00:33:29.120 --> 00:33:31.150
equal to the speed
along the curve
00:33:31.150 --> 00:33:32.860
that the particle
had at that point.
00:33:32.860 --> 00:33:35.340
And that's precisely
what dr/dt has.
00:33:35.340 --> 00:33:37.090
In other words,
what we have done
00:33:37.090 --> 00:33:40.890
is given a self-contained
mathematical derivation
00:33:40.890 --> 00:33:44.260
as to why the derivative
of the position vector r,
00:33:44.260 --> 00:33:46.690
with respect to time,
should physically
00:33:46.690 --> 00:33:49.060
be called the velocity vector.
00:33:49.060 --> 00:33:52.260
And then again, analogously
to ordinary physics,
00:33:52.260 --> 00:33:56.320
if v is the velocity vector, we
define the acceleration vector
00:33:56.320 --> 00:33:58.910
to be the derivative of the
velocity vector with respect
00:33:58.910 --> 00:33:59.860
to time.
00:33:59.860 --> 00:34:01.920
Now I was originally
going to close over here,
00:34:01.920 --> 00:34:03.570
but my feeling is
that this seems
00:34:03.570 --> 00:34:05.580
a little bit abstract
to you, so maybe we
00:34:05.580 --> 00:34:07.070
should take a
couple more minutes
00:34:07.070 --> 00:34:09.670
and do a specific problem.
00:34:09.670 --> 00:34:12.960
Let's take the particle that
moves along the path whose
00:34:12.960 --> 00:34:16.850
parametric equations are
y equals t cubed plus 1,
00:34:16.850 --> 00:34:18.570
and x equals t squared.
00:34:18.570 --> 00:34:21.595
In other words, notice that
for a given value of t-- t
00:34:21.595 --> 00:34:24.000
is a scalar-- for
a given value of t,
00:34:24.000 --> 00:34:26.090
I can figure out
where the particle is
00:34:26.090 --> 00:34:27.750
at any time along the curve.
00:34:27.750 --> 00:34:34.690
For example, when the time is
2, when t is 2, x is 4, y is 9.
00:34:34.690 --> 00:34:37.710
So at t equals 2, the particle
is at the point 4 comma
00:34:37.710 --> 00:34:39.520
9, et cetera.
00:34:39.520 --> 00:34:41.219
Notice that to
understand this problem,
00:34:41.219 --> 00:34:42.900
one does not need vectors.
00:34:42.900 --> 00:34:48.560
But if one knows vectors, one
introduces the radius vector r.
00:34:48.560 --> 00:34:49.949
What is the radius vector?
00:34:49.949 --> 00:34:52.880
Its i component
is the value of x,
00:34:52.880 --> 00:34:55.480
and its j component
is the value of y.
00:34:55.480 --> 00:34:58.470
So the radius vector
r is t squared i
00:34:58.470 --> 00:35:00.980
plus t cubed plus 1 j.
00:35:00.980 --> 00:35:04.230
Now, by these results,
I can very quickly
00:35:04.230 --> 00:35:07.195
compute both the velocity and
the acceleration of my particle
00:35:07.195 --> 00:35:08.410
at any time.
00:35:08.410 --> 00:35:10.670
Namely, to find the
velocity, I just
00:35:10.670 --> 00:35:12.621
differentiate this
with respect to t.
00:35:12.621 --> 00:35:13.870
This is going to give me what?
00:35:13.870 --> 00:35:16.130
The derivative of
t squared is 2t.
00:35:16.130 --> 00:35:20.730
The derivative of t cubed
plus 1 is 3 t squared.
00:35:20.730 --> 00:35:26.460
So my velocity vector is
just 2t*i plus 3 t squared j.
00:35:26.460 --> 00:35:28.500
Now to get the
acceleration vector,
00:35:28.500 --> 00:35:30.940
I simply have to differentiate
the velocity vector
00:35:30.940 --> 00:35:32.980
with respect to time.
00:35:32.980 --> 00:35:34.500
And I get what?
00:35:34.500 --> 00:35:37.130
2i plus 6t*j.
00:35:37.130 --> 00:35:40.200
And I now have the
acceleration, the velocity,
00:35:40.200 --> 00:35:44.195
and, by the way, the position
of my particle at any time t.
00:35:44.195 --> 00:35:45.570
In particular,
since I've already
00:35:45.570 --> 00:35:47.525
computed that the
particle is at the point
00:35:47.525 --> 00:35:50.500
4 comma 9 when t
is 2, let's carry
00:35:50.500 --> 00:35:53.370
through the rest of our
investigation when t is 2.
00:35:53.370 --> 00:35:56.980
When t is 2, notice
that in vector form,
00:35:56.980 --> 00:36:05.280
r is equal to 4i plus 9j;
v is equal to 4i plus 12j;
00:36:05.280 --> 00:36:12.100
and a is equal to 2i plus 12j.
00:36:12.100 --> 00:36:15.030
Pictorially-- and by the way,
to show you how to get this,
00:36:15.030 --> 00:36:17.630
all I'm saying here is
that if we come back
00:36:17.630 --> 00:36:22.900
here recalling that t cubed is
t squared to the 3/2 power--
00:36:22.900 --> 00:36:27.010
this simply says that y is
equal to x to the 3/2 plus 1.
00:36:27.010 --> 00:36:28.756
So I drew the path
in just so that you
00:36:28.756 --> 00:36:30.890
can get an idea of what's
going on over here.
00:36:30.890 --> 00:36:32.690
All I'm saying is
when somebody says,
00:36:32.690 --> 00:36:36.090
where is the particle at time
t equals 2, it's at the point
00:36:36.090 --> 00:36:37.620
4 comma 9.
00:36:37.620 --> 00:36:39.320
And notice the
correlation again.
00:36:39.320 --> 00:36:43.650
The point 4 comma 9 is the
point at which the vector 4i
00:36:43.650 --> 00:36:45.700
plus 9j terminates.
00:36:45.700 --> 00:36:49.390
Because you see, that
vector originates at (0, 0).
00:36:49.390 --> 00:36:52.630
I can compute the velocity
vector-- namely what?
00:36:52.630 --> 00:36:57.070
The velocity vector has its
slope equal to 3-- 12 over 4.
00:36:57.070 --> 00:36:59.280
And its magnitude is
the square root of 4
00:36:59.280 --> 00:37:01.080
squared plus 12 squared.
00:37:01.080 --> 00:37:04.380
That's the square root of 160,
which is roughly 12 and a half.
00:37:04.380 --> 00:37:07.300
I can now draw that
velocity vector to scale.
00:37:07.300 --> 00:37:10.530
In a similar way, the slope
of the acceleration vector
00:37:10.530 --> 00:37:13.740
at that point is 12
over 2, which is 6.
00:37:13.740 --> 00:37:15.640
So the slope is 6.
00:37:15.640 --> 00:37:17.720
That's a pretty steep line here.
00:37:17.720 --> 00:37:19.110
And the magnitude is what?
00:37:19.110 --> 00:37:22.590
The square root of 4 plus
144-- the square root
00:37:22.590 --> 00:37:25.400
of 148, which I just
call approximately 12,
00:37:25.400 --> 00:37:26.730
to give you a rough idea.
00:37:26.730 --> 00:37:29.260
I can draw this
into scale as well.
00:37:29.260 --> 00:37:33.950
In other words, in just
one short overview lesson,
00:37:33.950 --> 00:37:36.210
notice that we have
introduced what
00:37:36.210 --> 00:37:39.440
we mean by vector functions,
what we mean by limits.
00:37:39.440 --> 00:37:42.850
We've inherited the entire
structure of part one.
00:37:42.850 --> 00:37:45.500
I can now introduce
a position vector,
00:37:45.500 --> 00:37:47.320
differentiate it
with respect to time
00:37:47.320 --> 00:37:49.640
to find velocity
and acceleration,
00:37:49.640 --> 00:37:52.620
and I'm in business,
being able to solve
00:37:52.620 --> 00:37:55.270
vector problems in the
plane-- kinematics problems,
00:37:55.270 --> 00:37:57.270
if you will.
00:37:57.270 --> 00:38:00.160
We're going to continue on in
this vein for the remainder
00:38:00.160 --> 00:38:00.820
of this block.
00:38:00.820 --> 00:38:03.300
We have other coordinate
systems to talk about.
00:38:03.300 --> 00:38:05.500
Next time, I'm
going to talk about
00:38:05.500 --> 00:38:09.690
tangential and normal components
of vectors and the like.
00:38:09.690 --> 00:38:12.860
But all the time, the theory
that we're talking about
00:38:12.860 --> 00:38:13.940
is the same.
00:38:13.940 --> 00:38:17.470
Namely, we are given
motion in a plane.
00:38:17.470 --> 00:38:20.250
And we can now, in terms
of vector calculus,
00:38:20.250 --> 00:38:23.570
study that motion
very, very effectively.
00:38:23.570 --> 00:38:25.840
At any rate, until
next time, goodbye.
00:38:29.230 --> 00:38:31.600
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publication of this video
00:38:31.600 --> 00:38:36.480
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:38:36.480 --> 00:38:40.650
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